Partitioning of solutes in multiphase Ti Al alloys

Transcription

1 Partitioning of solutes in multiphase Ti Al alloys R. Benedek, A. van de Walle, S. S.A. Gerstl, M. Asta, and D. N. Seidman Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA C. Woodward Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA and Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright Patterson AFB, Ohio , USA Received 9 September 2004; revised manuscript received 1 December 2004; published 4 March 2005 First-principles calculations based on a plane-wave pseudopotential method, as implemented in the VASP code, are presented for the formation energies of several transition-metal and non-transition-metal dopants in Ti Al alloys. Substitution for either Ti or Al in -TiAl, 2 -Ti 3 Al, Ti 2 AlC, and Ti 3 AlC are considered. Calculated zero-temperature defect formation energies exhibit clear trends as a function of the periodic-table column of transition metal solutes. Early transition metals in TiAl prefer the Ti sublattice, but this preference gradually shifts to the Al sublattice for late transition metals; the Ti sublattice is preferred by all transition metal solutes in Ti 3 Al. Partitioning of solutes to Ti 3 Al is predicted for mid-period transition elements, and to TiAl for early and late transition elements. A simple Ising model treatment demonstrates the plausibility of these trends, which are in excellent overall agreement with experiment. The influence of temperature on formation energies is examined with a cluster expansion for the binary TiAl alloys and a low temperature expansion for dilute ternary alloys. Results for Nb-doped alloys provide insight into the relative sensitivity of solute partitioning to individual contributions to the free energy. Whereas the calculated formation energy of Nb substitution at zero temperature favors partitioning to 2 -Ti 3 Al, temperature-dependent contributions to the formation free energy, evaluated at 1075 K, favor partitioning to -TiAl, in agreement with experiment. DOI: /PhysRevB PACS number s : Dk, g I. INTRODUCTION Light weight, high strength, and corrosion resistance make multiphase TiAl alloys 1 attractive materials for hightemperature structural applications, particularly in the automotive and aerospace industries. 2 In spite of these intrinsic advantages, issues such as room-temperature brittleness, 3 only moderate creep resistance, 4,5 the development of optimal anticorrosion coatings, 6 optimal processing methods, 7 and stability in low ph environments 8 must be addressed before TiAl can be put into wider service. Although the tantalizing prospect of the replacement of Ni-based superalloys by TiAl in high-pressure turbine components now seems unlikely, 9 titanium aluminides are nevertheless expected to continue to be technologically important structural alloys. Dopants are often introduced into TiAl-alloy prototypes to improve their properties and performance. The selection of micro-alloying components has been guided primarily by empiricism. The detailed distribution of these dopants, as well as their influence on microscopic mechanisms that bear on materials properties, is poorly understood. Several types of solute behavior may be envisaged. Some solutes, such as C, 10,11 N, 10 B, and Si, form precipitates, whereas others form solid solutions in the host phases, from which they may preferentially segregate to defects and the heterophase interfaces in the system. Accurate information on the distribution of the solutes derived from simulation would be valuable for the future design and optimization of titanium-aluminide-based materials. The focus of this article is on bulk solute behavior. Some of our preliminary work on pristine interfaces of TiAl with carbides was published recently. 12 Formation energies for point defects, either intrinsic or extrinsic, in metallic systems are typically much larger than thermal energies k B T, and calculated defect energies at zero temperature are often sufficient for most purposes. In the case of Ti Al alloys, however, the effect of temperature on defect energies merits careful attention. For example, the differences in formation energy for a solute in different phases, or on different sublattices in a given phase, are generally much smaller than individual formation energies and may be comparable to k B T in some cases. 13 Moreover, the phase boundaries for the two-phase, 2 region of interest shift appreciably at elevated temperatures, driven by chemical potential changes that can also dramatically alter the relevant defect energies. Several applications of first-principles methods to predict point-defect concentrations in intermetallic compounds have appeared In these treatments, point-defect energy parameters derived from first-principles calculations are combined with Bragg-Williams lattice-gas model analyses of configurational entropy to predict the equilibrium concentrations of defects as a function of stoichiometry and temperature. Along similar lines, the site preferences for selected solutes in -TiAl have been predicted by combining firstprinciples calculations with a multi-component lattice-gas model. 13 Such treatments have thus far been restricted to a single alloy phase, and the consequences of a two-phase equilibrium for defect properties have not been explored. Many aspects of the analysis of defects in single-phase systems carry over to two-phase systems; in the latter, however, we require the chemical potentials of Ti and Al appropriate to compositions within the two-phase region. The chemical potentials in this region are, of course, closely related to the equilibrium-phase-diagram boundaries /2005/71 9 / /$ The American Physical Society

2 BENEDEK et al. Several simulations of the phase diagram of binary Ti Al have been published. Asta and co-workers used the cluster variation method to study phase stability for hcp and fcc phases. 17 More recently, van de Walle and co-workers reevaluated the Ti-rich region of the Ti Al phase diagram using Monte Carlo simulation based on a cluster expansion, 18 and included vibrational and electronic as well as configurational entropy. In this paper we investigate solute partitioning among bulk titanium aluminide phases -TiAl and 2 -Ti 3 Al and carbide-precipitate phases Ti 2 AlC and Ti 3 AlC. We focus particularly on the two-phase region, in which TiAl and Ti 3 Al are in equilibrium, which is the composition range of greatest practical interest. 1 We calculate the relevant energy parameters for solutes in these phases within the generalizedgradient approximation of local-density-functional theory LDFT-GGA. The solutes treated include most elements from transition groups IIIA to VIIIA, as well as B and Si. Considerable simplification of the theory occurs in the limits of dilute solute concentrations and zero temperature. Although these limits are not relevant to most technological applications of Ti Al alloys, the corresponding predictions represent a useful benchmark, and most of the results presented in this article will pertain to the zero-temperature dilute-solute limit. As touched upon earlier, however, a more comprehensive perspective is only possible with an understanding of how defect properties are modified at elevated temperatures. Therefore, we give considerable attention in this article to the calculation of the influence of temperature on solute formation energies. In our finite temperature calculations, however, we continue to resort to the dilute solute limit. The effects of solute-solute interactions and soluteinduced shifts in the phase boundaries, for example, are thereby excluded. Although such effects are not necessarily insignificant for practical dopant levels that may individually extend to several percent, and in the presence of several different solutes, it appears logical to address the dilute limit first. We employ a cluster expansion method 18 to calculate the free energy of the Ti Al binary alloy as a function of host composition and temperature. This approach enables phase boundaries and chemical potentials of the binary system to be determined. The effects of ternary solute elements are treated in the dilute limit with a low-temperature expansion. Nb, a commonly employed dopant with a wide solubility range, is selected as a representative solute in TiAl. 19 Owing to the extensive calculations required to evaluate the temperature-dependent contributions to the solute formation free energy, only Nb is considered here. In our low-temperature calculations, we find that, for group VA solutes and at least one group VIA solute, the difference between formation energies on the Ti and Al sublattices of -TiAl is of the order of a couple of tenths of an ev. Similarly, the difference between formation energies in the 2 phase and in the phase is small for several solutes. At low temperatures, Nb is predicted to favor the 2 phase. However, at elevated temperatures, Nb is predicted to switch preference to the phase, in agreement with experiment. In general, when the difference between the formation energies in the two phases is relatively small of the order of a tenth of an ev, thermal corrections to the formation free energies cannot automatically be assumed insignificant. The calculated formation energies at zero temperature for transition metals in the different phases show pronounced trends albeit with some scatter across the periodic table. For example, the sublattice preference of solutes in TiAl shifts from the Ti sublattice in the early transition metals to the Al sublattice in late transition elements, whereas the Ti sublattice is preferred uniformly for solutes in Ti 3 Al. Further, solutes in the middle of a transition series partition to Ti 3 Al, whereas only early and late transition metals prefer TiAl. The effects of temperature on the formation free energies are too weak to significantly perturb these trends. Some of the trends are captured by a simple Ising-type interaction model, which lends additional credibility to the results. Close agreement of the predicted trends with experiment is found, where comparison is possible. II. SOLUTE FORMATION ENERGIES AND FREE ENERGIES The partitioning of dilute concentrations of solutes among the phases of the alloy at thermal equilibrium is determined by their relative formation energies or free energies in those phases. We evaluate in this work the formation energies of solute species of interest in the host titanium-aluminide phases -TiAl and 2 -Ti 3 Al, and in carbide precipitate phases. The formation energy of species i on sublattice j of phase may be written as 20 E F i, j, = E i, j, N, 1 where E is the cohesive energy change upon substitution of an i atom for a host atom on the j sublattice of phase, and N accounts for the chemical potentials of atoms added or removed from thermodynamic reservoirs during the formation of the defect. Thus, for a substitutional solute, N = i k j, where k j denotes the atomic species on sublattice j of phase. E i, j, is evaluated with the lattice surrounding the solute relaxed to equilibrium. This formulation of the solute formation energy is applicable at zero temperature, and at ideal host stoichiometry, when the atomic species, k j, that occupies each sublattice is unique. At elevated temperatures, the formulation should be generalized to the case of partial occupancy, and corrected for lattice expansion and thermal disorder, as well as shifts in the phase boundaries of -TiAl and 2 -Ti 3 Al and host chemical potentials. The formation energy, E F i, j,,t, is thus temperature dependent. Vibrational entropy contributions TS vib F i, j,,t to the free energy 21,22 also become significant at elevated temperatures. Formally, the equilibrium concentration of impurity species i is related to the derivative of the thermodynamic potential with respect to its chemical potential. 20 In the dilute limit, this concentration is exponentially related to the thermodynamic free energy of formation, F i, j,,t = E F i, j,,t TS vib F i, j,,t : c i, j,,t = exp F i, j,,t /k B T. 2 Although Eq. 2 is well established for stoichiometric com

3 PARTITIONING OF SOLUTES IN MULTIPHASE Ti Al pounds, its applicability to alloys with appreciable deviations from stoichiometry has apparently not been discussed. The Appendix of this paper shows that Eq. 2 holds for the case of dilute ternary additions to binary compounds with concentrated antisite disorder, provided the defect formation free energy is defined as a configurational average. III. NUMERICAL RESULTS First-principles calculations of total energies are performed within the generalized-gradient-approximation GGA 23 of density functional theory, as implemented in the VASP code. 24 The calculations employ a plane-wave basis combined with the ultrasoft pseudopotential formalism. 25 Calculations are performed for four different phases, -TiAl and 2 -Ti 3 Al, as well as the carbides Ti 2 AlC Htype and Ti 3 AlC P-type. Special k-point sets based on Monkhorst-Pack 26 indices 888 are employed. In all of the calculations for cells with defects intrinsic point defects or solute atoms, the internal coordinates are relaxed to equilibrium, as mentioned above. In principle, of course, the cell dimensions lattice constants also relax in the presence of point defects. This correction, however, is often negligible in practice for periodic supercells of metallic systems with a few dozen atoms or more. A test calculation was performed for an Al vacancy in a 32-atom -TiAl cell; similar volume-relaxation energies are anticipated for the 64-atom cell employed for the 2 phase. The equilibrium volume in the presence of the Al vacancy in TiAl decreased by about 1%, relative to that for the defect-free crystal, and the total energy of the cell decreased by about ev, relative to that obtained at the defect-free crystal equilibrium volume. The influence of volume relaxation on point defect energies for solutes is expected to be of similar magnitude to that for vacancies, or smaller. Energy changes of this magnitude are small relative to the scale of most of the energies calculated in this work; the effects of such volume-relaxation energies on trends is expected to be especially small, because of cancellations. The results presented in the remainder of this article correspond to cells in which the volumes are fixed at the equilibrium values for the ideal stoichiometries. A. Host material lattice constants Calculations for tetragonal L1 0 -TiAl utilized a 32-atom cell, which corresponds to doubling the lattice constants of the conventional face-centered-tetragonal unit cell. Equilibrium lattice constants in Angstrom units a=7.96/2=3.98 experiment: 4.005, and c=8.182/2=4.09 experiment: 4.07, were obtained. Calculations for hexagonal D Ti 3 Al were based on a 64-atom cell, obtained by doubling the hexagonal-layer lattice constants, as well as the lattice constant perpendicular to the layers. The predicted equilibrium lattice constants are a=11.50/2=5.75 experiment: 5.78 and c=9.304/2=4.652 experiment: Calculations for Ti 2 AlC, a hexagonal system, were based on a 64- atom cell obtained by doubling the hexagonal-layer lattice constants and the c-axis lattice constant. The predicted equilibrium lattice constants are a=6.14/2=3.07 experiment: 3.04 and c=27.384/2=13.69 experiment: Calculations for Ti 3 AlC, a cubic perovskite, were based on a 40- atom cell obtained by doubling the cubic lattice constant. The predicted equilibrium lattice constants are a = 8.36/ 2 =4.18 experiment: For the first three of these structures, a is slightly underestimated and c and therefore c/a is slightly overestimated, which is often found for layered or tetragonal systems within the LDFT-GGA. B. Chemical potentials The defect formation energy, Eq. 1, depends on the chemical potentials of the host elements Ti and Al. Of particular interest are compositions within the two-phase region in which TiAl and Ti 3 Al are in equilibrium. If the formation energies of intrinsic point defects, i.e., vacancies, interstitials, and antisite atoms, are positive, the chemical potentials at zero temperature Ti and Al satisfy the relations and Ti + Al = TiAl, 3 Ti + Al = Ti 3 Al, 4 where TiAl and Ti 3 Al are the cohesive energies, per formula unit, of the two phases. VASP supercell calculations yield TiAl = ev and Ti 3 Al = ev from which we obtain Ti = ev and Al = ev. The assumption of positive intrinsic defect formation energies may be verified a posteriori, based on these chemical potentials. C. Intrinsic point defect energies Intrinsic defect formation energies can be expressed in terms of E i, j,, the difference between the total energy of a computational unit cell with a vacancy or an antisite atom, and the energy of a defect-free reference cell. Thus, E F v, j,,0 = E v, j, j 5 is the vacancy formation energy on sublattice j of phase at zero temperature, where j is the chemical potential of the atomic species on sublattice j. Similarly, E F j, j,,0 = E i, j, i + j 6 is the formation energy of an antisite atom substitution onto sublattice j of an atom of the species on sublattice j, where j j. Numerical results for both vacancies and antisite atoms are listed in Table I. Since the defect energies are all positive, the ideal compound stoichiometries of TiAl and Ti 3 Al are stable at low temperatures. This is in contrast to the behavior at elevated temperatures, 27 at which the phase boundary of the 2 phase corresponds approximately to Ti 2 Al rather than Ti 3 Al. We observe in Table I that the formation energies of vacancies on either sublattice of TiAl or Ti 3 Al are considerably larger than those of antisite atoms, which indicates that de

4 BENEDEK et al. TABLE I. Formation energies of intrinsic defects in two-phase TiAl,Ti 3 Al region. Phase E F V Ti E F V Al E F a Ti E F a Al TiAl Ti 3 Al viations from stoichiometry at most temperatures are accommodated primarily by antisite atoms, and the number of vacancies in thermal equilibrium is quite small. D. Solute formation energies The solute formation energy E F i, j,,0 = E i, j, i + j, 7 where i refers to the solute species. The phases under consideration are labeled = for TiAl, = 2 for Ti 3 Al, =H for Ti 2 AlC, and =P for Ti 3 AlC. The relative formation energy of a given solute i in phases 1 and 2, E F i = E F i, j 1, 1,0 E F i, j 2, 2,0, 8 is independent of the solute chemical potential, i. We require only that its value be consistent with the assumption of diluteness. Incidentally, knowledge of the solute chemical potential is necessary to predict absolute solubility in a given phase, 28 which is not addressed in this article. Once the host chemical potentials are known, the chemical potential for carbon can be obtained from the total energy of the carbide: C = Ti 2 AlC 2 Ti Al, 9 where Ti 2 AlC is the total energy of the H-phase per formula unit. An almost identical value of C is obtained from the energy of the P-phase, Ti 3 AlC. We note, incidentally, that Eqs. 3, 4, and 9 do not imply vanishing compound heats of formation, because the chemical potentials Ti and Al correspond to the two-phase region rather than to the respective elemental materials. 1. Solute sublattice preference in host phases The differences between the formation energies of solutes on the j 1 =Al and j 2 =Ti sublattices in the = and = 2 phases, E F i,0 = E F i, j 1,,0 E F i, j 2,,0, 10 are plotted in Fig. 1. The ordinate represents E F i and the abscissa the periodic-table column of the solute. For groups where more than one member is considered e.g., Cr, Mo, and W in group VIA, the members are offset from each other in intervals of 0.1 for example, Cr: 5.9, Mo: 6.0, W: 6.1. A preference for the Ti sublattice in TiAl occurs for early transition metal solutes, whereas the later transition metals increasingly prefer the Al sublattice. This variation across the periodic table is in contrast to solute behavior in FIG. 1. Difference between the formation energies for solute substitution on the Al sublattice and on the Ti sublattice of TiAl filled circles and Ti 3 Al filled diamonds. Results are shown for selected solutes in the transition metal series IIIB VIIIB. The nontransition elements Si and B not represented in the plot strongly favor the Al sublattice in both phases. The abscissa value is shifted slightly to enable members of the same group to be distinguished, e.g., Cr: 5.9, Mo: 6.0, W: 6.1. The trends in the data indicate that the Al sublattice in TiAl is increasingly preferred for the later transition groups. The crossover between Ti preference and Al preference occurs roughly between groups VB and VIB. Transition metal solutes in Ti 3 Al all prefer the Ti sublattice. Ti 3 Al, in which the Ti sublattice is preferred by all solutes. Although the trends exhibited in Fig. 1 are clear, a fair amount of scatter is found for the TiAl phase, particularly for different members of groups VA and VIA. For these groups, the formation energies on the Ti and Al sublattices of TiAl are fairly comparable. Solutes with small E F i 0.1 ev, approximately are shared between the two sublattices. 13 The lack of a strong sublattice preference makes the temperaturedependent contributions to the formation free energies of particular interest for these groups. Temperature-dependent contributions to the formation free energy of Nb are presented below. Results for the nontransition element solutes B and Si, analogous to those plotted in Fig. 1, show that substitution on the Al sublattice is preferred by both in either aluminide phase. The energy differences between the two sublattices for these solutes is in the range 0.8 to 1.0 ev. 2. Solute partitioning We consider next the relative formation energies of solutes in TiAl and Ti 3 Al. We plot in Fig. 2 the difference E F 2 i = E F i, j 2, 2,0 E F i, j,,0, 11 where j 2 denotes the preferred sublattice of solute i in the 2 -phase and j the preferred sublattice in the phase. A negative value of E F 2 i signifies that the 2 phase is preferred. Almost all solutes considered prefer the 2 phase, the

5 PARTITIONING OF SOLUTES IN MULTIPHASE Ti Al FIG. 2. Difference between the formation energies for solute substitution in Ti 3 Al and TiAl for selected solutes in the transition metal series IIIB VIIIB. The formation energy that corresponds to the preferred sublattice cf. Fig. 1 for each phase is employed. Apart from a few early Y and late Mn,Ru transition metals, a distinct preference for the Ti 3 Al phase is observed. exceptions being the early transition metal Y and the late transition metals Mn, Fe, and Ru, which have only a slight preference for the phase. Groups VB and VIB show a particularly strong preference for 2. Nontransition elements are predicted to partition to the 2 phase: E F 2 i = 0.3 for B, and 0.6 ev for Si. 3. Solute formation energies in carbides Transition metal solutes in the H- and P-type carbides substitute primarily on the Ti sublattice. An exception is the group IIIB element Y in the H-type carbide, which substitutes on the Al sublattice. In Fig. 3 we plot the difference between the lowest formation energy of solutes in the H-type carbides and the lowest formation energy in the host phases. Negative values indicates a preference for carbide over host. Of the two carbides, the H-type carbide is preferred by all transition metal solutes except Sc. In Fig. 3, only Y shows a negative relative formation energy; we note that Sc has a relative formation energy close to zero, with respect to the P-type carbide not shown in plot. The trend of later transition metal groups is to favor the host phases over carbides by increasing amounts. E. Temperature dependence Since service temperatures of interest for titanium aluminides extend to C about 60% to 70% of the melting temperature and processing temperatures are somewhat higher, the behavior of defect properties in these temperature ranges would be of interest. Nb was selected for investigation, because a relatively marginal phase preference for this solute was found in zero temperature calculations, and finite-temperature corrections may therefore be required FIG. 3. The difference between the formation energies for solute substitution in the preferred carbide Ti 2 AlC or Ti 3 AlC and in the aluminide host either Ti 3 Al or TiAl, whichever has the lower energy. to get qualitatively correct results. Results will be reported for a temperature of 1075 K. Let us consider the free energy of formation for a solute atom, F i, j,,t. At zero temperature, F i, j,,0 =E F i, j,,0. We express the temperature correction to the formation free energy, F i, j,,t = F i, j,,t F i, j,,0, as F i, j,,t = F h i, j,,t + F t i, j,,t + F i, j,,t + F stoi i, j,,t. F h i, j,,t is the vibrational free energy of defect formation obtained within the harmonic approximation, whereas F t i, j,,t is the contribution of thermal expansion to the vibrational free energy. F i, j,,t represents the change in defect formation energy that results from the temperature dependence of the chemical potentials within the two-phase region. Finally, F stoi i, j,,t is the shift in defect formation energy due to deviations from ideal stoichiometry. The calculation of each term will be described in turn. 1. Harmonic vibrational free energy of defect formation F h i,j,,t First-principles lattice dynamics calculations are especially demanding numerically for the relatively lowsymmetry configurations associated with defects. To reduce the computational requirements, we rely on the transferable length-dependent force constant methodology. 21,22 Each chemical bond is characterized by an analytical stiffnesslength relationship, whose parameters are fitted to firstprinciples calculations. The Al Nb and Ti Nb parameters were fitted to calculations for titanium aluminide supercells with a Nb substitution, whereas those for the Al Al, Al Ti, and Ti Ti bonds were obtained from supercells without substitutions. 18 Based on the stiffness-length relationships, the force constants for other atomic configurations such as supercells

6 BENEDEK et al. TABLE II. Temperature-dependent contributions to free energy of formation of Nb solutes in TiAl and Ti 3 Al at 1075 K see Sec. III E for details. Phase Sublattice F 0 F h T F t T F T F stoi T F T TiAl Al TiAl Ti Ti 3 Al Al Ti 3 Al Ti with embedded point defects can be estimated. The resultant force constants are input to a Born von Kármán lattice dynamics e.g., Ref. 21 analysis of phonon spectra, in terms of which vibrational free energy within the harmonic approximation is obtained see the fourth column of Table II. 2. Quasi-harmonic free energy of defect formation F h i,j,,t + F t i,j,,t Thermal expansion may be incorporated into the formulation by considering phonon frequencies to be volume dependent, the so-called quasi-harmonic approximation e.g., Ref. 21. The equilibrium volume and the corresponding free energy at a given temperature is obtained by minimizing the sum of the elastic strain energy of the expanded lattice and the volume-dependent vibrational free energy. The quasi-harmonic treatment yields the sum F h i, j,,t + F t i, j,,t, and results listed in the fifth column of Table II for F t i, j,,t are obtained by subtracting the harmonic free energy from the quasiharmonic free energy. Since the force constants employed in the harmonic calculations are parametrized by bond length, implementation of the quasi-harmonic approximation required no additional force constant calculations. To calculate the elastic energy cost, the bulk moduli of the 2 and phases were fitted to total energy calculations for lattice parameters up to 2% above the equilibrium value. For simplicity, the effect of solutes on the bulk moduli was neglected, as was the slight anisotropy in the thermal expansion of the 2 and phases. 3. Chemical-potential-induced shift F i,j,,t The formation free energies of Nb solutes in the 2 and phases within the two-phase region depend on the host chemical potentials in the temperature range of interest. As mentioned earlier, the chemical potential of Nb is constrained by the assumption of diluteness, but is otherwise arbitrary. For our present purpose, only the temperature dependence of Al Ti within the two-phase region is required. The small deviation of the -phase boundary from stoichiometry can be accurately accounted for with a lowtemperature expansion 20 based on the antisite energies given in Table I to determine the free energy of the phase as a function of Al content. The antisite formation energies were corrected for the effect of lattice vibrations, within the quasiharmonic approximation, by procedures similar to those outlined above. To model the 2 phase, whose boundaries deviate markedly from stoichiometry at elevated temperatures, we employ a cluster expansion 29 in conjunction with Monte Carlo simulations. The cluster expansion parameters were fitted to a database generated with first-principles calculations using VASP. These calculations 18 also incorporate lattice vibrational effects, modeled via the transferable force constant scheme and within the quasi-harmonic approximation. Most of the employed codes are incorporated in the ATAT program package. 30 Using the calculated free energies of the and the 2 phases at T = 1075 K, we find a two-phase equilibrium at Ti Al = ev; for comparison, Ti Al = ev at T=0 K. The effect of this chemical potential shift on the formation free energy is given in the sixth column of Table II. 4. Stoichiometric effect F stoi i,j,,t Finally, let us consider the change in the energy of defect formation associated with deviations from the ideal stoichiometries at finite temperatures. In principle, a ternary cluster expansion that incorporates Nb as well as Ti and Al would be useful in this context, but is beyond the scope of the present work, and a more approximate approach was used. Since stoichiometric effects are expected to be most significant for 2 phase, we will focus on that phase. In the 64-atom cell employed in calculations for the D0 19 structure, we consider an arrangement with 42 Ti and 22 Al atoms. This composition is close to the observed stoichiometry at high temperatures and can be realized by creating six Al antisite atoms in the ideal Ti 48 Al 16 cell used to represent the low-temperature 2 system. Among the large number of possible arrangements of six antisite Al atoms in the 64-atom cell, we select for computation an arrangement in which the antisite atoms are relatively well separated. For this configuration, prospective sites for substitution on the Ti sublattice may have either zero, one, or two Al antisite atom neighbors, with approximately equal combinatorial weight. Explicit calculations for configurations with either zero or two Al antisite atom neighbors of a Nb solute indicate that F stoi i, j,,t is about 0.3 ev for Nb substitution on the Ti sublattice see the seventh column of Table II. Since Nb favors the Ti sublattice, the stoichiometric correction for the Al sublattice is not calculated. While this result accounts for the most important energetic contribution to F stoi i, j,,t, it neglects several higher-order effects. For example, the presence of a Nb impurity may modify the distribution of surrounding Al and Ti atoms. The treatment of such contributions would be feasible with a ternary cluster expansion

7 PARTITIONING OF SOLUTES IN MULTIPHASE Ti Al TABLE III. The calculated and experimental values of binary alloy properties for composition Ti 1 x Al x. The results of present calculations are listed in the third column. The lattice constants and alloy formation energies H are calculated for the ideal stoichiometries x th = for the 2 phase at zero temperature. The monovacancy formation energies are calculated for off-stoichiometric compositions, as described in the text. The bottom two rows give the phase boundary between the two phase region and the 2 phase, and between the two phase region and the phase. The last column cites references for experimental results. Phase Property Theory x th Experiment x expt Reference a Å VC Ref. 47 c Å VC 2 a Å VC 2 c Å VC H ev/at KH Ref H ev/at KH E V F ev 1.82, E V F ev B Ref E V F ev B 2 E V F ev B 2 x 2 T=1075 K O Ref. 27 x T=1075 K O IV. DISCUSSION A. Intrinsic host material properties Comparisons of calculated host-material properties with experiment see Table III corroborate the accuracy of the LDFT-GGA framework. Such comparisons support our confidence in predictions of solute properties. Comparisons of the predicted solute properties directly with experiment will also be presented; however, such comparisons are more ambiguous than those for the pure host phases. 1. Lattice constants Predicted lattice constants for the compounds under investigation are generally within 1% of experimental values cf. Sec. III A which is typical of the LDFT-GGA. Predicted compound formation energies, H, of TiAl and Ti 3 Al overshoot experimental values 31 by about 10%, consistent with earlier first-principles treatments of this property Point defect energies Predicted intrinsic-defect energies at zero temperature, assuming ideal stoichiometry, are given in Table I. Experimental values of vacancy formation energies derived from positron annihilation spectroscopy, given in Table III, are in the range of 1.55 ev for 2 -Ti 77 Al 23 and 1.41 ev for -Ti 48.5 Al The present theoretical predictions are somewhat higher, particularly for Ti 3 Al. The discrepancies between the values listed in Table I and experiment are most likely related primarily to the assumptions of ideal stoichiometry and zero temperature. For the phase, however, deviations from stoichiometry are relatively small. Using a lattice gas model 13 and fixing Ti Al = ev at 1075 K, in accordance with results presented in the previous section, we find the phase has a composition of Ti Al The corresponding calculated vacancy formation energies on the Al and Ti sublattices are 1.82 and 1.91 ev, respectively. For the composition appropriate to the positron annihilation experiments of Brossmann et al. 33 Ti 48.5 Al 51.5, at which vacancies occur predominantly on the Ti sublattice, we predict a Ti vacancy formation energy of 1.41 ev, in excellent agreement with experiment as well as previous first principles calculations 35 see Table III. We note, however, that the present analysis does not address the severe electron-density depletion near vacancies that make them stronger perturbations to the host crystal than solute atoms, in general. Accordingly, additional corrections to the LDFT-GGA formulation may be necessary before critical comparisons of calculated vacancy formation free energies with experiment can be made. 36 These complications are not pursued here, however, since our focus is on solutes. 3. Phase boundaries Phase-boundary compositions of the two-phase region in which the TiAl and Ti 3 Al phases coexist may be extracted from the the temperature dependence of the chemicalpotential difference Al Ti discussed in Sec. III E. The consistency of these predictions with the experimental phase diagram is a significant test. At 1075 K, the predicted phase boundaries are at 31% and 47% Al, respectively, for Ti 3 Al and TiAl. The corresponding experimental values 27 are approximately 35% and 48% Al. The TiAl phase boundary is seen to be in good agreement with experiment, and its closeness to the equiatomic limit justifies our application of a low-temperature expansion for this phase. On the other hand, a discrepancy of about 4% appears for the Ti 3 Al phase boundary. The precise location of the Ti 3 Al phase boundary is highly sensitive to small errors in the calculated free en

8 BENEDEK et al. ergies, because the concentration dependence of the free energy of this phase exhibits only weak curvature, which is partly a consequence of the small Al antisite energy see Table I. Thus, a relatively large error for this phase boundary is perhaps not surprising. A more detailed discussion of the errors inherent in cluster-expansion-based calculations of phase boundaries is given by Ghosh et al. 37 B. Ising-model description of formation energy trends The trends in the formation energies of substitutional defects in the and 2 phases cf. Figs. 1 and 2 must follow from trends in such properties as band filling and electronegativity. Rather than attempting to resolve precisely which property is responsible, we explore instead a formulation of total energy based on a nearest-neighbor pseudo-spin Ising model, in which the pseudo-spin represents phenomenologically the variation in the solute chemistry across the periodic table. Within the nearest neighbor Ising model, the energy E of a binary TiuAl alloy either the or the 2 phase is written as E = i i + J i j, i,j 12 where the spinlike occupation variables i take the value +1 Al or 1 Ti depending on the identity of the atom located at site i. The sums are taken, respectively, over all lattice sites i and over all neighboring pairs of lattice sites i, j. The parameter is the chemical potential difference Al Ti between Al and Ti within the two-phase region cf. Sec. III B, while J specifies the strength of the interatomic interactions in phase = 2 or. For the fcc lattice associated with the phase, the interaction parameter was taken from the nearest neighbor interaction obtained in Ref. 38 J =0.064 ev. For the 2 phase, the nearest-neighbor interaction parameter was taken from the more accurate thermodynamic model used in Sec. III, with in-layer and out-of-layer interactions on the hexagonal lattice averaged J 2 =0.045 ev. Although the parameters, J 2, and J are not all independent within a nearest-neighbor model, we treat as an adjustable parameter because of its sensitivity to the long-range interactions neglected in the Ising model. The Ising Hamiltonian, Eq. 12, can be extended in a formal way to multicomponent systems by including additional parameters; 29 however, a simpler treatment is adopted here. Since Ti and Al almost bracket the range of the periodic table in which the transition elements reside, it is reasonable, from a heuristic viewpoint, to characterize the chemical behavior of a given transition element via a linear interpolation between Ti and Al, in terms of the periodic table column, c. Accordingly, solute atoms are assigned a pseudo-spin i = 1+2 c 4 /9, which reduces to 1 for Ti and +1 for Al. These fractional pseudo-spin variables are then used in conjunction with Eq. 12 to predict the properties plotted in Figs. 1 and 2. The calculation of these defect formation energy differences is performed in the dilute limit by considering a single isolated impurity, and the detailed impurity distribution therefore need not be specified. The adjustable parameter was set to match the crossing of the horizontal FIG. 4. The predictions of solute formation energies based on simplified Ising model, Eq. 12. Panel a is to be compared with Fig. 1 and panel b with Fig. 2. axis of the TiAl data in Fig. 1. The resultant value, = 0.38 ev/atom, is within 25% of the value obtained directly from the nearest-neighbor model 0.51 ev/atom. The model predictions, analogous to the first-principles calculations plotted in Figs. 1 and 2, are shown in Figs. 4 a and 4 b. The Ising model captures the most salient features of the trends observed in the formation energies calculated with first-principles methods. First, as seen in Figs. 1 and 4 a, transition-metal solutes always prefer the Ti sublattice of the Ti 3 Al phase, while, for the TiAl phase, the solute preference shifts from the Ti to the Al sublattice as the transition series is traversed. Second, as seen in Figs. 2 and 4 b, solutes tend to prefer the 2 phase, particularly in the middle of a transition element series. Further insight into this behavior can be gained by recognizing that Ti Al is an ordering system for which it is favorable to maximize the number of unlike bonds i.e., nearest-neighbor bonds between distinct chemical species. Ti sites in ordered stoichiometric Ti 3 Al, with eight like and four unlike nearest neighbors, would prefer more unlike neighbors, which can be achieved by substitution of atoms with i 1 on the Ti sublattice. Solute substitution on the Al sublattice, whose sites already have all unlike neighbors, confers no such advantage, and substitution on the Ti sublattice of the Ti 3 Al phase is therefore preferred. Ordered L1 0 TiAl has the maximum number eight of unlike bonds possible on an fcc lattice, so that solutes with

9 PARTITIONING OF SOLUTES IN MULTIPHASE Ti Al values of i intermediate between those of Ti and Al effectively decrease the number of unlike bonds. The energy cost of substitution is minimized when the more aluminumlike solutes i.e., with larger c reside on the Al sublattice, and the more titaniumlike solutes i.e., with smaller c reside on the Ti sublattice. This switch in preference is at the origin of the sign change in Figs. 1 and 4 a and also explains the presence of a minimum in the middle of the transition metal sequence in Figs. 2 and 4 b, where substitutions on either sublattice of the L1 0 structure are relatively unfavorable. We have seen that the Ising model predictions, based on one adjustable parameter, are consistent with the most significant features of the trends exhibited in Figs. 1 and 2. The correspondence between the first-principles calculations and the Ising model is, of course, not exact in all details for example, the best fit line to results for Ti 3 Al in Fig. 1 appears horizontal, whereas the line in Fig. 4 a shows a small negative slope. Extensions of the Ising model would presumably bring the model and the first-principles results into closer correspondence. C. Solute sublattice preference in the phase The sublattice partitioning of ternary elements in a binary host can be determined experimentally by the atom-locationchanneling-enhanced-microanalysis technique ALCHEMI, 39 statistical ALCHEMI, 40 as well as by atomprobe field ion microscopy 41 and x-ray diffraction. 42 Ideally, ALCHEMI yields the fraction f Ti or f Al of a given solute species on the Ti or Al sublattice. For large values of E F Fig. 1 we can approximate 1/f Ti =1+w exp E F i,t, 13 where w is the ratio of Ti to Al atoms in the stoichiometric compounds, if deviations of the host phase from ideal stoichiometry, and the consequent antisite host-atom populations, may be ignored. Although this approximation may be justified for -TiAl in some cases, it is unlikely to be accurate for the 2 phase. For single-phase -TiAl, a lattice gas model was developed to predict the fractional site occupancy for a given composition and temperature 13 when E F is small. For comparison with experiment, we perform calculations for a nominal composition of Ti 51 ual M 1, where M is the ternary element, which is close to the boundary of the binary 2 + phase field for temperatures in the range K. 43 Results for the various ternary additions are plotted in Fig. 5 together with the statistical ALCHEMI results of Rossouw et al. 40 The crossover that occurs between predominant Ti sublattice occupancy of early transition elements and predominant Al sublattice occupancy of late transition elements is clear in both theory and experiment. The plotted results are in close agreement with previous LDFT calculations on a smaller set of solutes. 13 We note that in both Fig. 5 and Fig. 6, discussed below, a larger discrepancy appears for Mn than for most other solutes. The results for Mn were not significantly changed by including spin polarization. A higher-level approximation than LDFT-GGA may be required to achieve greater accuracy for Mn. FIG. 5. The predicted open circles and measured filled circles partitioning of transition-metal solutes in -TiAl with composition Ti 51 Al M 1 at 1473 K. The ordinate represents the fraction of solutes that reside on the Ti sublattice. D. Solute partitioning: Experiment The ratio of solute concentration in TiAl to that in Ti 3 Al may be written c i, j, 2,T /c i, j,,t = w p exp E F 2 i, 14 assuming ideal stoichiometries, where w p is the sublattice factor. For stoichiometric Ti 3 Al and TiAl, the sublattice factor w p =2/3, if the solutes in Ti 3 Al are assumed to reside on the Ti sublattice, as suggested by the results in Fig. 1. For an 2 -phase stoichiometry close to Ti 2 Al, a slightly larger sublattice factor would be appropriate. In Fig. 6, numerical results for transition metal solutes are plotted, using w p =1, along with experimental results given by Rossouw et al. 40 Both ALCHEMI measurements 40 and three-dimensional atom probe microscopy observations 44,45 indicate that Zr and Nb prefer the phase, whereas other transition-metal solutes FIG. 6. Predicted filled circles and measured diamonds ratio of solute concentration in the phase to that in the 2 phase

10 BENEDEK et al. Hf, Cr, Ta, Mn, V, W, and Mo preferentially partition to the 2 phase. Thus, an apparent discrepancy exists between experiment and the zero-temperature predictions shown in Fig. 2, in which both Nb and Zr exhibit lower formation energies in Ti 3 Al than TiAl, by 0.25 and 0.08 ev, respectively. We now consider whether elevated-temperature corrections are sufficiently large to reverse the sign of the formation energy difference between the phases. E. Temperature-dependent contributions to formation free energies In Sec. III E, several types of temperature-dependent contributions to the solute formation free energies were calculated for Nb at a reference temperature of 1075 K see Table II. We consider the dilute limit, and the quantities F i, j,,0 and F i, j,,t in Table II therefore do not reflect contributions associated with the Nb chemical potential, as mentioned previously. The temperature corrections due to lattice vibrations, thermal expansion, and host-chemical-potential shift columns 4 6 in Table II are fairly small and do not reverse the zerotemperature preference for the Ti sublattice of the 2 phase; however, the correction due to the deviation of the Ti 3 Al phase from ideal stoichiometry seventh column is relatively larger. This term was estimated only for the Ti sublattice, which is preferred by Nb in each phase cf. Fig. 1. Comparing the results in the first and last columns of the table, the formation energy difference F (Nb, Ti, 2,T) F Nb,Ti,,T is seen to shift from at 0 K to at 1075 K, indicating that temperature corrections do appear to reverse the Nb solute phase preference from 2 at zero temperature to at 1075 K. Our predictions at 1075 K are therefore consistent with experimental observation. Although numerical calculations of the temperature-dependent shift in formation energies were not performed for solutes other than Nb, it is reasonable to expect the sign of the shift to be the same for most solutes. Therefore, we anticipate that similar calculations to those for Nb would show that the sign of E F Nb, 2 E F Nb, for Zr shifts to positive values at elevated temperatures, in accord with experiment. F. Solute formation energies in carbides One of the motivations for this work was to identify solutes that segregate to carbide precipitates, or their interfaces with the host matrix, and consequently enhance their stability against coarsening. The trends exhibited by the results plotted in Fig. 3 suggest that the early groups IIIB and IVB transition metals are the best candidates for such segregation, although even for these early transition elements, the zerotemperature calculations yield formation energies higher in the carbide than in the matrix for all solutes except Y. Ta favors the carbide over TiAl, but not over Ti 3 Al. In most of these comparisons, the solute formation energy in the H-type carbide was employed; transition-metal solutes have a lower formation energy in the H-type than the P-type carbide, with Sc the sole exception. Zr showed essentially identical formation energies in H- and P-type carbides. Based on these results, Y would appear to be the best candidate for a solute that partitions to carbide precipitates. Rossouw et al., 40 however, note that Y, as well as La, exhibits negligible solute solubility in the host, but instead forms Al 2 X precipitates. In zero-temperature simulations, we find that Y does show a strong attraction for Ti 2 AlC/TiAl interfaces, 46 and Y migration to such interfaces therefore remains a possibility. Overall, the present simulations do not suggest a strong candidate for segregation to carbide precipitates, although when effects of temperature and stoichiometry, which are excluded from our calculations, as well as interactions with carbide-host interfaces are taken into account, different behavior may emerge. V. SUMMARY AND CONCLUSIONS Formation energy calculations for dilute substitutional solutes in TiAl and Ti 3 Al, as well as in the H- and P-type carbide precipitates, have been performed with firstprinciples methods, using the VASP code. The selected solutes include transition metals from the three transition periods, and nontransition elements B and Si, which represents virtually the entire set of dopants that are commonly used in Ti Al alloys. Most of the calculations correspond to zero temperature, but corrections to the solute formation free energy appropriate to 800 C were calculated in the case of Nb dopants. One source of corrections to F larger than k B T in absolute magnitude is associated with the stoichiometry shift of the 2 phase at elevated temperatures. Other corrections of the order of k B T are identified. Although the corrections to formation free energies at 800 C are of the order of tenths of an ev, and therefore not insignificant, the main trends exhibited by the zero-temperature solute formation energies as a function of periodic table group or number of outershell electrons appear to be borne out in experiments that measure atomic configurations representative of high temperatures. The experimental probes that provide the most critical tests of our calculations are the Statistical ALCHEMI method 40 and three-dimensional atom-probe microscopy. 44,45 Our calculations indicate that all transition metal solutes prefer the Ti sublattice of Ti 3 Al. In TiAl, however, the preference of early transition-metal groups for the Ti sublattice crosses over to a preference for the Al sublattice by later transition groups. The zero-temperature calculations predict the crossover to occur in the vicinity of groups V and VI, which is roughly consistent with experiment. Our calculations predict that the solutes in the middle of the transition series partition strongly to Ti 3 Al, whereas those at either end partition roughly equally between the two phases, or slightly favor TiAl. Experiment agrees with this overall trend, but shows discrepancies for individual solutes, such as Nb, which may be removed with the inclusion of thermal corrections to the solute formation free energy. The zero-temperature formation energy calculations do not identify any solute that would be likely to partition to carbide precipitates. The best prospects for such partitioning, however, are group IIIB and IVB elements